# Philosophical aspects of the foundations of mathematics

Let us complete our initiation to the foundations of mathematics by more philosophical aspects : how, independently of our time, the mathematical realm is structured by a growing block flow of its own "time".
• First, 1.A to 1.C will explain this "time" as affecting model theory ;
• then, its role in set theory (clarifying the distinction of sets among classes) will be explored in 1.D, 2.A, 2.B, 2.C.
These complements are not needed to continue with Part 2 (2.1 to 2.10 except for small remarks in 2.2, 2.7 and 2.10), Part 3 and more, while 2.A-2.C assumes both Part 1 down to 1.D and Part 2 down to 2.7.

## 1.A. Time in model theory

### The time order between interpretations of expressions

The interpretations of expressions of a theory T in a model M depend on each other, thus come as calculated after each other. This time order follows the construction order from sub-expressions to expressions containing them.
For example, to make sense of the formula xy+x=3, the free variables x and y must take values first; then, xy takes a value, obtained by multiplying them. From this, xy+x takes a value, and then the whole formula (xy+x=3) takes a Boolean value, depending on the values of x and y. Finally, taking for example the ground formula ∀x, ∃y, xy+x=3, its Boolean value (which is false in the world of real numbers), «is calculated from» those taken by the previous formula for all possible values of x and y, and therefore comes after them.

The interpretations of a finite list of expressions may be gathered by a single other expression, taking them as sub-expressions. This big expression is interpreted after them all, but still belongs to the same theory.
Now for a single expression to handle an infinite set of expressions (such as the range of all expressions of T, or just all terms or all statements), these expressions must be treated as objects (values of a variable). If T is a foundational theory, it can define (or construct) a system looking like this, so that, in any standard model of T (in a sense we shall specify later), this definition will designate an exact copy of this set of expressions.

However, the systematic interpretation of all expressions of T in M cannot fit any definition by a single expression of T interpreted in the same M. Namely, this forms a part of the combined system [T,M] beyond M, to be described by a one-model theory (1MT) which, even if it can be developed from T, anyway requires another interpretation, at a meta level over M.

### The infinite time between models

Without trying to formalize the model theory MT of first-order logic (which will be approached in the next sections) let us sketch a classification of its components (mixing notions, structures and axioms) into parts according to what they describe. Most issues are unchanged by restricting consideration to an 1MT, with model a single system [T,M] of a theory T with a model M, when these T and M (specified by additional axioms of 1MT) are big enough to roughly contain any theory and any system. But some issues may even be unchanged with simpler choices of T and M.
• A Theory theory TT (resp. a one-theory theory 1TT) itself made of successive parts which roughly follow the layers of the theory T it describes :
• A description of "abstract types" and "symbols" ;
• The described notions of "expressions" and "statements" ;
• 1TT can be formalized from TT by adding unary predicate symbols τ, L, X which select the classes of components of T (its types, symbols and axioms) from some sets of "all possible ones";
• Proof theory can be first expressed in TT by the notion of "proof" for logically valid statements; then the notions of "contradiction", "proof" and "theorem" of a theory T can be defined from τ, L, X using the "logical validity" of statements made of disjunctions and implications of other statements.
• A Systems Theory describing M by interpreting there the types and structure symbols of T given from 1TT (if the types list and the language of T are finite and explicitly given then the role of its one-system theory may be played by T itself; otherwise it involves the meta-notions of "types" and "structures").
• The description of the interpretations (attribution of values) of all expressions of T in M, for any values of their free variables; thus also, interpretations of all statements. This is needed to express MT (the truth of all axioms of T in M) if the axioms list is infinite.
This last part describes a part of [T,M] which is determined by the combination of both systems T and M but not directly contained in them : it is built after them.

### The metaphor of the usual time

I can speak of «what I meant at that time»: it has a sense if that past saying had one, as I got that meaning and I remember it. But mentioning «what I mean» in isolation, would not itself inform on what it is, as it might be anything, and becomes absurd in a phrase that modifies or contradicts this meaning («the opposite of what I mean»). Mentioning «what I will tell tomorrow», even if I already knew what I will say, would not suffice to already provide its meaning either: in case I will mention «what I told about yesterday» (thus now) it would make a vicious circle; but even if the form of my future saying ensured that its meaning will exist tomorrow, this would still not provide it today. Regardless my speculations, the actual meaning of expressions yet to be uttered will only arise in their time, from the context that will come.
By lack of interest to describe phrases without their meaning, we'd rather focus on previously uttered expressions, while just "living" the present ones and ignoring future ones. So, my current universe of the past that I can describe today, includes the one I could describe yesterday, but also my yesterday's comments about it and their meaning. I can thus describe today things outside the universe I could describe yesterday. Meanwhile I neither learned to speak Martian nor acquired a new transcendental intelligence, but the same language applies to a broader universe with new objects. As these new objects are of the same kinds as the old ones, my universe of today may look similar to that of yesterday; but from one universe to another, the same expressions can take different meanings.

Like historians, each mathematical theory can only «at any given time» describe a system of past mathematical objects. Its interpretation in this system, «happens» forming a mathematical present outside this realm (beyond this past). Then, describing this act of interpretation, means expanding the scope of our descriptions : the model [T,M] of 1MT, encompassing the interpretations of all expressions of T in the present system M of past objects, is the next realm of the past, coming once the infinite totality of current interpretations (in M) of expressions of T becomes past.

### The strength hierarchy of theories

While these successive models are separated by "infinite times", they form an endless succession, reflected by an endless hierarchy between the theories which respectively describe them. This hierarchy will be referred to as a comparison of strength of theories (this forms a preorder). Namely, a theory A is called stronger than a theory B if (a copy of) B can be found as contained in A or a possible development of A; they are as strong if this also goes vice-versa. Indeed, developments are mere "finite moves" neglected by the concept of strength which aims to report "infinite moves" only. (Other definitions of strength order, often but maybe not always equivalent, will come in 2.C).

Many strengths will be represented by versions of set theory, thus letting us call "universes" these successive models. So, any set theory being meant as describing some universe of «all mathematical objects», this merely is at any time the current «everything», made of our past, while this description itself forms something else beyond this «everything».

### Strengthening axioms of set theory

While we shall focus on set theories accepting other notions than sets (as announced in 1.4), the difference with traditional set theories (with only sets as objects) can be ignored, as any worthy set theory formalized in our way is as strong as one with only sets, and similarly vice versa.
Our set theories, beyond their common list of basic, "necessary" symbols and axioms (2.1 and 2.2) will mainly differ by strength, according to their choices of optional strengthening axioms (sometimes coming with primitive symbols), whose role will be further commented in 1.D and 2.C. The main strengthening axioms are :
• Infinity (4.4) : there exists an infinite set, or equivalently a set ℕ of all natural numbers;
• The Specification schema amounts to generalizing the use of the set builder to unary predicates A defined using open quantifiers. But this amounts to recognizing the universe and other classes as kinds of sets (though not objects of any single kind) and hides the possible dependence of the result on the universe (range of all objects), which will often be conceived as variable (2.A). These oddities are usually limited by rejecting the set builder notation as inappropriate, leaving this as an axiom schema
(∀A)∀SetE, ∃SetF, ∀x, xF ⇔ (xEA(x))
• Out of the scope of this introduction, the Collection schema implies the Replacement schema, which implies the Specification schema, with possible converses depending on other axioms.
• Powerset (2.7)

### The main foundational theories

As a simplified introduction, here are some of the main foundational theories (all first-order theories, even "second-order arithmetic"), ordered by increasing strength (while infinities of other strengths also exist between and beyond them).
• Let us call Finite Objects theories (FOT) these 3 theories which are of the same strength:
• First-order arithmetic (Z1), reducing induction to an axiom schema by second-order universal elimination.
• Finite Set Theory (FST), with Specification schema and negation of Infinity
• Theory theory (TT).
• Model theory (MT), which can interpret all expressions in all its definable countable systems (= made of ℕ with any defined structures); as strong as a version of set theory from which most of ordinary mathematics (analysis...) can be derived ;
• Second-order arithmetic (Z2), formalizable as set theory with Infinity and Specification (does Replacement make it stronger ?), or (almost equivalently but in a different formalism) Infinity and the powerset of only ℕ;
• Mc Lane set theory, with Infinity and Powerset, is the comfortable one for most needs;
• Zermelo set theory is slightly stronger, with Infinity, Powerset and Specification.
• Zermelo-Fraenkel set theory (ZF) is much stronger, with Infinity, Powerset and Replacement; it implies Collection.
The hardest part of Gödel's proof of his famous incompleteness theorems, was to develop TT from Z1, so that the incompleteness results first proven for TT also affect Z1. This difficulty can be skipped by focusing on TT and FST, ignoring Z1. Developing TT from FST is easy (once TT is formalized), but developing either from Z1 is harder. A solution is to develop the "sets only" version of FST from Z1 by defining the BIT predicate (to serve as ∈) and proving its basic properties ; the difficulty to do so can be skipped by accepting these as primitive.

 Set theory and Foundations of Mathematics 1. First foundations of mathematics 1.9. Axioms and proofs 1.10. Quantifiers 1.11. Second-order quantifiers ⇦ Philosophical aspects 1.A. Time in model theory ⇨1.B. Truth undefinability 2. Set theory - 3. Algebra - 4. Arithmetic - 5. Second-order foundations
Other languages:
FR : Temps en théorie des modèles